Thetanulls of cyclic curves of small genus

Emma Previato, Tanush Shaska, G. S. Wijesiri

I-AIMS paper 2007-3 · Published in Albanian J. Math. 1 (2007), 253–270 · 2007

We study relations among the classical thetanulls of cyclic curves, namely curves 𝒳 (of genus g(𝒳) > 1) with an automorphism σ such that σ generates a normal subgroup of the group G of automorphisms, and g(𝒳/⟨σ⟩) = 0. Relations between thetanulls and branch points of the projection are the object of much classical work, especially for hyperelliptic curves, and of recent work, in the cyclic case. We determine the curves of genus 2 and 3 in the locus ℳ_g(G, C) for all G that have a normal subgroup ⟨σ⟩ as above, and all possible signatures C, via relations among their thetanulls.

MSC classes: 14H32, 14H37, 14K25

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References

1. G. Cardona, J. Quer, Field of moduli and field of definition for curves of genus 2. Computational aspects of algebraic curves, 71–83, Lecture Notes Ser. Comput., 13, World Sci. Publ., Hackensack, NJ, 2005.
2. A. Krazer, Lehrbuch der Thetafunctionen, Chelsea, New York, (1970).
3. R. Kuhn, Curves of genus 2 with split Jacobian. Trans. Amer. Math. Soc. 307 (1988), no. 1, 41–49.
4. H.F. Baker, Abelian Function, Abel’s theorem and the allied theory of theta functions, (1897).
5. K. Magaard, T. Shaska, S. Shpectorov, H. Vlklein, The locus of curves with prescribed automorphism group. Communications in arithmetic fundamental groups (Kyoto, 1999/2001). S¯urikaisekikenky¯usho K¯oky¯uroku No. 1267 (2002), 112–141.
6. D. Mumford, Tata lectures on theta. II. Jacobian theta functions and differential equations. With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. Progress in Mathematics, 43. Birkhuser Boston, Inc., Boston, MA, 1984.
7. D. Mumford, Tata lectures on theta. I. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. Progress in Mathematics, 28. Birkhuser Boston, Inc., Boston, MA, 1983. xiii+235 pp.
8. A.Nakayashiki, On the Thomae formula for ZN curves, Publ. Res. Inst. Math. Sci., vol 33 (1997), no. 6, pg. 987–1015.
9. T. Shaska, Curves of genus 2 with (N, N) decomposable Jacobians, J. Symbolic Comput., vol. 31, Nr. 5, 2001, 603–617.
10. Algebraic curves and their applications http://www.albmath.org/algcurves/
11. H. Shiga, On the representation of the Picard modular function by θ constants. I, II., Publ. Res. Inst. Math. Sci., vol. 24, (1988), no. 3, pg. 311–360.
12. H.E. Rauch and H.M.Farkas, Theta functions with applications to Riemann surfaces, Williams and Wilkins, Baltimore, 1974.

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